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Correspondences in Arakelov Geometry and Applications to the Case of Hecke Operators on Modular Curves

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Correspondences in Arakelov Geometry and Applications to the Case of Hecke Operators on Modular Curves CORRESPONDENCES IN ARAKELOV GEOMETRY AND APPLICATIONS TO THE CASE OF HECKE OPERATORS ON MODULAR CURVES RICARDO MENARES Abstract. In the context of arithmetic surfaces, Bost defined a generalized Arithmetic 2 Chow Group (ACG) using the Sobolev space L1. We study the behavior of these groups under pull-back and push-forward and we prove a projection formula. We use these results to define an action of the Hecke operators on the ACG of modular curves and to show that they are self-adjoint with respect to the arithmetic intersection product. The decomposition of the ACG in eigencomponents which follows allows us to define new numerical invariants, which are refined versions of the self-intersection of the dualizing sheaf. Using the Gross-Zagier formula and a calculation due independently to Bost and K¨uhnwe compute these invariants in terms of special values of L series. On the other hand, we obtain a proof of the fact that Hecke correspondences acting on the Jacobian of the modular curves are self-adjoint with respect to the N´eron-Tate height pairing. Contents 1. Introduction1 2 2. Functoriality in L1 5 2 2.1. The space L1 and related spaces5 2 2.2. Pull-back and push-forward in the space L1 6 2 2.3. Pull-back and push-forward of L1-Green functions 12 3. Pull-back and push-forward in Arakelov theory 16 2 3.1. The L1-arithmetic Chow group, the admissible arithmetic Chow group and intersection theory 16 3.2. Functoriality with respect to pull-back and push-forward 18 4. Hecke correspondences on modular curves 21 4.1. Integral models of the modular curves X0(N) and Hecke correspondences 21 4.2. Arithmetic Chow group of X0(N) 23 4.3. Arithmetic Chow group of X0(N) with real coefficients 28 4.4. Self-intersection and refined self-intersection of the dualising sheaf 30 2 5. Appendix: cohomology of L1 spaces on compact Riemann surfaces 34 References 36 arXiv:0911.0546v3 [math.NT] 6 Mar 2013 1. Introduction Let X be an arithmetic surface, that is a projective, regular, 2-Krull dimensional scheme over Spec(Z). Such an object arises naturally as a model over the integers of a smooth algebraic curve defined over a number field. Arakelov devised a formalism on X analogous to the classical intersection theory on an algebraic surface defined over a field ([Ara74]). The basic object of this formalism is a compactified divisor, that is a pair (D; g), where D is a Weil divisor on X and g is a real-valued distribution on the compact Riemann surface X = X (C). Here, g is a Green function for D, subject to the following condition. 1 Suppose for simplicity that the genus of X is positive, and take an orthonormal basis !1;:::;!n of the space of holomorphic 1-forms (where we take this space to be endowed i R with the inner product (α; β) 7! 2 X α ^ β). The canonical Arakelov 2-form is defined by n i X µ := ! ^ ! : Ar n j j j=1 This is a C1 volume form on X. It is independent of the choice of the orthonormal basis f!1;:::;!ng. We impose that g satisfies the equation of currents c (1.1) dd g + δD(C) = deg D(C) µAr: If this condition is satisfied, then g is said to be admissible. We also impose the normal- R ization X gµAr = 0. The key analytic ingredient needed to define an intersection number from two com- pactified divisors (D1; g1) and (D2; g2) is the star product c (1.2) g1 ∗ g2 = g1(dd g2 + δD2(C)) + g2δD1(C): This is a well defined current if D1(C) and D2(C) do not have points in common. The R integral g1 ∗ g2 is bilinear and symmetric in (g1; g2). X (C) In Gillet and Soul´e'ssubsequent treatment of Arakelov's theory [GS90], the admissi- bility condition on g is dropped. More precisely, g is a real-valued C1 function on the P complement of the support of D(C) = P aP [P ]. It is required that for every point P in this support, there is a local expansion of the form 2 (1.3) g = −aP log jθP (·)j + b; 1 with θP a local chart vanishing at P and b a C function. Such a distribution is called a Green current for D. Equation (1.1) implies the local expansion (1.3), making Gillet and Soul´e'stheory a generalization of Arakelov's theory. The star product (1.2) between normalized Green currents retains the symmetry and bilinearity properties. In this paper we study the lack of functoriality problem in Arakelov theory. That is, let ' : X!Y be a finite morphism between arithmetic surfaces, and (D; g) a compactified divisor on X . If the induced holomorphic covering 'C : X (C) !Y(C) is ramified, then the pair ('∗D; '∗g) induced from the the push-forward is not a compactified divisor on Y, in either the Arakelov or the Gillet-Soul´esense. The reason is purely analytic in nature; '∗µAr has singularities at the branching points, hence it is not the Arakelov canonical form on Y(C). Furthermore, '∗g is not a Green current for '∗D, because the push-forward operation does not preserve C1 functions. In this sense, the theories of Arakelov and Gillet-Soul´eare not functorial. In many situations, it is natural to consider correspondences on arithmetic surfaces. Unfortunately, the interaction with the intersection formalism is excluded by this phenomenon. 2 Bost brought to this circle of ideas the Sobolev space L1, which consists of distributions in L2 whose derivatives also belong to L2. He proposed a generalization of Gillet-Soul´e's theory for arithmetic surfaces ([Bos99]). In this framework, g is a real-valued element of 2 L1 away from the support of D(C) and for every point P in this support, there is a local 2 2 expansion similar to (1.3) but where b is in L1. Such an objet is called a L1-Green function. 2 The integral of the star product (1.2) between L1-Green functions is well defined. This 2 theory is optimal in the sense that L1 is the biggest space such that a useful (integral 2 of a) star product can be defined. Bost's theory also improves on the geometrical part, because it doest not require the scheme X to be regular but only integral and normal. We drop the regularity in the definition of arithmetic surface in what follows. 1 2 Let us write CH[(X ); <; > to denote the L1 arithmetic Chow group defined in [Bos99]. In this paper, we prove that these groups have a good functoriality. More precisely, we provide a proof of the following Theorem 1.1. Let ' : X!Y be a generically finite morphism between integral, normal, arithmetic surfaces. (1) The natural definitions of pull-back and push-forward induce homomorphisms '∗ : CH[1(Y) −! CH[1(X ) 1 1 '∗ : CH[(X ) −! CH[(Y): (2) Projection formula: for all x 2 CH[1(X ) and y 2 CH[1(Y), we have that ∗ <' y; x> = <y; '∗x>: The main difficulty in proving this theorem is to control the L2 norm of '∗f around 2 the ramification points, where f is L1. This is achieved by establishing a Poincar´e-like inequality (Lemma 2.10 in the main text), which we present as a variant of the Hardy inequality. On the other hand, the push-forward operation turns out to be easier to handle. We point out that Theorem 1.1 was known to Bost before our investigation started ([Bos99], p. 245, [Bos98]), though he did not publish any details. The method described in the paragraph above, different from Bost's approach, was found independently (cf. Remark 2.12). 2 We also consider a generalization of Arakelov's condition (1.1). Let L−1 X (C) be the @ @ space of 2-currents which are locally of the form (f + @z h1 + @z¯h2)dz ^ dz¯, where f; h1; h2 are L2. Let µ be a real-valued element of L2 X ( ) such that R µ = 1. Given a −1 C X (C) 2 Weil divisor D on X , we call a µ-admissible Green function for D an L1-Green function g for D such that equation c dd g + δD(C) = deg D(C) µ 1 holds in the sense of currents. We denote by CH[(X )µ the corresponding µ-admissible arithmetic Chow group. We show that pull-back and push-forward are well-behaved with respect to this notion, as well as study how the reference 2-current changes. 2 2 Theorem 1.2. Let µ 2 L−1 X (C) and ν 2 L−1 Y(C) be real-valued 2-currents such that R µ = R ν = 1. The natural definitions of pull-back and push-forward induce X (C) Y(C) homomorphisms ∗ 1 1 ' : CH[(Y)ν −! CH[(X ) '∗ν deg ' 1 1 '∗ : CH[(X )µ −! CH[(Y)'∗µ: Here, the main issue is the question of the existence of such µ-admissible Green func- tions (cf. Proposition 3.2). Such a distibution always exists locally, a result coming from the theory of elliptic PDEs. This reduces our problem to patching together local solu- 2 tions to obtain a global one, which in turn is an exercise in L1-cohomology of Riemann surfaces. This exercise is probably well known to the experts, though for lack of suitable 3 reference in the literature, we collect in the appendix the necessary facts and deductions to complete it.
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